Provided by: libmath-planepath-perl_122-1_all 
NAME
Math::PlanePath::FlowsnakeCentres -- self-similar path of hexagon centres
SYNOPSIS
use Math::PlanePath::FlowsnakeCentres;
my $path = Math::PlanePath::FlowsnakeCentres->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path is a variation of the flowsnake curve by William Gosper which follows the
flowsnake tiling the same way but the centres of the hexagons instead of corners across.
The result is the same overall shape, but a symmetric base figure.
39----40 8
/ \
32----33 38----37 41 7
/ \ \ \
31----30 34----35----36 42 47 6
\ / / \
28----29 16----15 43 46 48--... 5
/ / \ \ \
27 22 17----18 14 44----45 4
/ / \ \ \
26 23 21----20----19 13 10 3
\ \ / / \
25----24 4---- 5 12----11 9 2
/ \ /
3---- 2 6---- 7---- 8 1
\
0---- 1 <- Y=0
-5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
The points are spread out on every second X coordinate to make little triangles with
integer coordinates, per "Triangular Lattice" in Math::PlanePath.
The base pattern is the seven points 0 to 6,
4---- 5
/ \
3---- 2 6---
\
0---- 1
This repeats at 7-fold increasing scale, with sub-sections rotated according to the edge
direction, and the 1, 2 and 6 sub-sections in reverse. Eg. N=7 to N=13 is the "1" part
taking the base figure in reverse and rotated so the end points towards the "2".
The next level can be seen at the midpoints of each such group, being
N=2,11,18,23,30,37,46.
---- 37
---- ---
30---- ---
| ---
| 46
|
| ----18
| ----- ---
23--- ---
---
--- 11
-----
2 ---
Arms
The optional "arms" parameter can give up to three copies of the curve, each advancing
successively. For example "arms=>3" is as follows. Notice the N=3*k points are the plain
curve, and N=3*k+1 and N=3*k+2 are rotated copies of it.
84---... 48----45 5
/ / \
81 66 51----54 42 4
/ / \ \ \
28----25 78 69 63----60----57 39 30 3
/ \ \ \ / / \
31----34 22 75----72 12----15 36----33 27 2
\ \ / \ /
40----37 19 4 9---- 6 18----21----24 1
/ / / \ \
43 58 16 7 1 0---- 3 77----80 <- Y=0
/ / \ \ \ / \
46 55 61 13----10 2 11 74----71 83 -1
\ \ \ / / \ \ \
49----52 64 73 5---- 8 14 65----68 86 -2
/ / \ / / /
... 67----70 76 20----17 62 53 ... -3
\ / / / / \
85----82----79 23 38 59----56 50 -4
/ / \ /
26 35 41----44----47 -5
\ \
29----32 -6
^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
As described in "Arms" in Math::PlanePath::Flowsnake the flowsnake essentially fills a
hexagonal shape with wiggly sides. For this Centres variation the start of each arm
corresponds to the centre of a little hexagon. The N=0 little hexagon is at the origin,
and the 1 and 2 beside and below,
^ / \ / \
\ \ / \
| \ | |
| 1 | 0--->
| | |
\ / \ /
\ / \ /
| |
| 2 |
| / |
/ /
v \ /
Like the main Flowsnake the sides of the arms mesh perfectly and three arms fill the
plane.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::FlowsnakeCentres->new ()"
Create and return a new path object.
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if
"$n < 0" then the return is an empty list.
Fractional positions give an X,Y position along a straight line between the integer
positions.
"($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
In the current code the returned range is exact, meaning $n_lo and $n_hi are the
smallest and biggest in the rectangle, but don't rely on that yet since finding the
exact range is a touch on the slow side. (The advantage of which though is that it
helps avoid very big ranges from a simple over-estimate.)
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 7**$level - 1)", or for multiple arms return "(0, $arms * 7**$level - 1)".
There are 7^level points in a level, or arms*7^level for multiple arms, numbered
starting from 0.
FORMULAS
N to X,Y
The "n_to_xy()" calculation follows Ed Schouten's method
<http://80386.nl/projects/flowsnake/>
breaking N into base-7 digits, applying reversals from high to low according to digits 1,
2, or 6, then applying rotation and position according to the resulting digits.
Unlike Ed's code, the path here starts from N=0 at the edge of the Gosper island shape and
for that reason doesn't cover the plane. An offset of N-2*7^21 and suitable X,Y offset
can be applied to get the same result.
X,Y to N
The "xy_to_n()" calculation also follows Ed Schouten's method. It's based on a nice
observation that the seven cells of the base figure can be identified from their X,Y
coordinates, and the centre of those seven cell figures then shrunk down a level to be a
unit apart, thus generating digits of N from low to high.
In triangular grid X,Y a remainder is formed
m = (x + 2*y) mod 7
Taking the base figure's N=0 at 0,0 the remainders are
4---- 6
/ \
1---- 3 5
\
0---- 2
The remainders are unchanged when the shape is moved by some multiple of the next level
X=5,Y=1 or the same at 120 degrees X=1,Y=3 or 240 degrees X=-4,Y=1. Those vectors all
have X+2*Y==0 mod 7.
From the m remainder an offset can be applied to move X,Y to the 0 position, leaving X,Y a
multiple of the next level vectors X=5,Y=1 etc. Those vectors can then be shrunk down
with
Xshrunk = (3*Y + 5*X) / 14
Yshrunk = (5*Y - X) / 14
This gives integers since 3*Y+5*X and 5*Y-X are always multiples of 14. For example the
N=35 point at X=2,Y=6 reduces to X = (3*6+5*2)/14 = 2 and Y = (5*6-2)/14 = 2, which is
then the "5" part of the base figure.
The remainders can be mapped to digits and then reversals and rotations applied, from high
to low, according to the edge orientation. Those steps can be combined in a single lookup
table with 6 states (three rotations, and each one forward or reverse).
For the main curve the reduction ends at 0,0. For the multi-arm form the second arm ends
to the right at -2,0 and the third below at -1,-1. Notice the modulo and shrink procedure
maps those three points back to themselves unchanged. The calculation can be done without
paying attention to which arms are supposed to be in use. On reaching one of the three
ends the "arm" is determined and the original X,Y can be rejected or accepted accordingly.
The key to this approach is that the base figure is symmetric around a central point, so
the tiling can be broken down first, and the rotations or reversals in the path applied
afterwards. Can it work on a non-symmetric base figure like the "across" style of the
main Flowsnake, or something like the "DragonCurve" for that matter?
SEE ALSO
Math::PlanePath, Math::PlanePath::Flowsnake, Math::PlanePath::GosperIslands
Math::PlanePath::KochCurve, Math::PlanePath::HilbertCurve, Math::PlanePath::PeanoCurve,
Math::PlanePath::ZOrderCurve
<http://80386.nl/projects/flowsnake/> -- Ed Schouten's code
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under the terms
of the GNU General Public License as published by the Free Software Foundation; either
version 3, or (at your option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with Math-
PlanePath. If not, see <http://www.gnu.org/licenses/>.